Abstract

The local-global similarity model of V1 proposes two transformations to derive
the organisation of orientation preference. Both these transforms are mappings
of the hemi-retinal image, and are achieved by known connectivity in the visual
system. The transformations are a global retinotopic mapping into the
supragranular layers and a tiling of the supragranular layers with multiple
local versions of the hemi-retinal image. The local mapping implies that the
lateral connections in lamina 4B preserve the topographic ordering of the
visual hemi-field. Since the local mapping of the supragranular layers produces
a simultaneous expansion and contraction of the current visual inputs, it
accounts for both the local response properties for orientation preference and
their global geometric organisation. The interaction of the two topographically
identical mappings is shown to predict the formation of patchy intrinsic
connections in the supragranular layers. The mappings described allow those
properties of the visual field which can be predicted on the basis of spatial
contiguity (texture, orientation, colour, contrast) to be available as response
properties over the entire visual field.

1 Introduction

This paper presents a model of the mammalian primary visual cortex. The model
assumes that the geometry of V1 serves to organise multiple response properties
into two cortical dimensions. The core proposition is the existence of two,
topographically identical mappings of the visual field to the primary visual
cortex, which define the geometrical organisation of orientation preference and
drive the formation of patchy connectivity in the supragranular layers.

The local-global similarity (LGS) model proposes that V1 is doubly-similar in
the following sense. Input from the eye reaches the supragranular layers of V1
via two mappings: a global mapping and a local mapping. Multiple copies of the
local mapping tile the area of V1. The global mapping supplies retinotopic
information to the upper layers. Each local mapping (via interactions with the
global mapping and other local maps) supplies additional response properties
such as orientation preference. The LGS model of the primary visual cortex
suggests there is a direct relationship between globally represented objects
(e.g. an oriented line) and various other response properties (e.g. orientation
preference) which have a local geometry.

The function of the LGS mapping is to provide each point in the retinotopic
image with a representation of the entire visual hemi-field. Each retinotopic
point is able to learn a `snapshot' of points in the visual image whose
activity tends to coincide with that retinotopic point. In the case of
orientation preference, the set of oriented lines passing through a retinotopic
point can be associated with that point.

2 Response Property Geometry of V1

The response properties of V1 are well catalogued. These properties have been
mapped through single-cell studies
(Hubel and Wiesel, 1968), metabolic
transport studies
(Tootell et al, 1988a; 1988b; 1988c)
and optical imaging of
the cortical surface
(Blasdel, 1992; Blasdel and Salama, 1986).
These studies
have shown that V1 exhibits a distinctive geometrical tiling
(Blasdel, 1992).
Swindale (1996) has described a set of canonical properties which a model of
the geometry of the primary visual cortex should take into account. These
properties include the spatial frequencies and organisation of: ocular
dominance bands, cytochrome oxidase (CO) blobs, singularities and
iso-orientation regions. To Swindale's list may be added: spatial frequency
selectivity, patchy intrinsic connectivity and cortical point spread.

Layer 4C of the macaque primary visual cortex has a strict retinotopic
organisation
(Blasdel and Fitzpatrick, 1984).
This layer receives the vast bulk
of inputs from the retina, via the thalamus
(Blasdel and Lund, 1983).
In the
macaque, each hemi-retina is distorted, through a complex logarithmic mapping,
to an almond shaped mapping in layer 4C of V1
(Tootell et al, 1988a). This
distortion is of fundamental importance to the functionality of the visual
recognition system, allowing computational simplification for operations such
as rotation and scaling in two dimensions
(Schwartz, 1980; Sheridan and Alexander, 1997).
Receptive field studies
(Hubel and Wiesel, 1977) reveal an
orderly, though less exact retinotopic organisation in the supragranular
layers. Receptive field sizes are also larger in the supragranular layers.

Neurons in the upper layers of the primary visual cortex are organised into
repeated units, roughly 800 um wide and 600 um high, called hypercolumns
(Hubel and Wiesel, 1968).
Each hypercolumn spans an entire range of orientation
tunings and a left and right ocular dominance set. Located along the centres of
ocular dominance bands are cytochrome-oxidase (CO) blobs. The blobs appear to
CO staining because of their higher metabolic activity
(Horton and Hubel, 1980),
and are responsive to colour, high contrast and low spatial frequencies.
Interblob regions are more selective for low contrast and high spatial
frequencies
(Tootell et al, 1988b; Tootell et al, 1988c).

The geometry of orientation preference reveals three predominant features:
singularities, linear zones and saddle regions
(Blasdel, 1992). Orientation
preference changes continuously around points, or singularities. These roughly
circular `pinwheels' traverse 180o
of all possible orientation preference.
Between adjacent singularities, running parallel to the ocular dominance bands,
are regions in which orientation preference changes slowly and continuously.
These regions are called linear zones. Other regions between singularities show
local minima of orientation preference in orthogonal directions: so-called
saddle points. Singularities surrounding saddle-points form mirror images of
each other, through both a vertical reflection line and a horizontal one. The
double reflections have the effect of allowing orientation selectivity to
change continuously across the edges singularities.

Singularities and CO blobs both tend to lie along the centres of ocular
dominance bands
(Bartfield and Grinvald, 1992; Livingstone and Hubel, 1984).
In
most cases CO blobs lie in between singularities, although in up to 15% of
cases singularities and CO blobs coincide
(Bartfield and Grinvald, 1992).
It is
not known whether the incidence of CO blob and singularity overlap varies over
the surface of V1, for example, whether it is more prevalent in the foveal
region.

3 Connectivity of Granular and Supragranular Layers

Anatomically, the mapping into layer 4C occurs via the Lateral Geniculate
Nucleus (LGN) in the thalamus
(Blasdel and Lund, 1983). In layer 4C, two
streams of inputs from the magnocellular and parvocellular layers of the LGN
are recombined to form an orderly retinotopic representation
(Blasdel and Fitzpatrick, 1984).
The inputs from the LGN project to layer 4C in discrete
blocks. Projections from the parvocellular layers of the LGN form dense
terminal boutons in lamina 4C.
The size of these dense terminating patches is
approximately the width of an ocular dominance column, 400 um. The terminating
input patches to 4C
lie exclusively within one or other ocular dominance band.
Magnocellular inputs project to 4C
in finer patches of approximately 100 um.

The supragranular layers receive little or no input directly from 4C, but
receive the bulk of their inputs via lamina 4A and 4B
(Blasdel et al, 1985).
Lamina 4C
projects to the supragranular layers largely via lamina 4A.
Lamina 4A is a major source of inputs to the supragranular layers, in the form
of diffusely spreading axons from 4A spiny stellates
(Blasdel et al, 1985; Yoshioka et al, 1994).
Lamina 4B receives most of its input from
4C. These
inputs in turn project strongly to the supragranular layers in narrow foci,
directly above each point in 4B. Some coarse fibres project into the
supragranular layers up to a distance of 375 um
(Blasdel, 1985).

Of particular interest for the model presented in this paper are the lateral
projections of lamina 4B spiny stellates within 4B itself. These efferent axons
project laterally up to 4.5 mm in the macaque
(Blasdel et al, 1985).
A similar
pattern of projecting fibres is found in the squirrel monkey
(Rockland and Lund, 1983).
The lateral connections show periodic accumulations of denser
terminal fibres every 375-400 um
(Blasdel et al, 1985; Rockland and Lund, 1983).
These patches of connections form a radial pattern with a similar
spatial periodicity to the CO blobs
(Rockland and Lund, 1983).
The fibres
within 4B extend further than any other class of intrinsic fibres within the
primary visual cortex. Only cells within layer 4B send out this class of long
range lateral fibres and, at mid- to long-range distances from the cell body,
all terminating boutons are within 4B itself. Most of the fibres within 4B are
preferentially horizontal, rather than vertical
(Rockland and Lund, 1983).

The set of connections within 4B is ideally suited to carry out the mapping,
proposed in this paper, of the retinotopic representation in the granular
layers to a local mapping in the supragranular layers. The chief requirement of
this mapping is that each coarse-grained (~ 400 x 300 um) retinotopic location
in lamina 4C connects to a point in every local map (~ 400 x 300 um) in the
supragranular layers. This mapping occurs through polysynaptic pathways within
lamina 4B.

Studies using injections of retrograde and antereograde tracer have revealed
underlying regularities in patchy intrinsic connections within the
supragranular layers of V1
(Blasdel et al, 1985; Bosking et al, 1997; Malach et
al, 1993; Rockland and Lund, 1983; Hubel and Wiesel).
These connections
traverse the grey matter parallel to the cortical surface in the supragranular
layers and project to discrete patches or targets, sometimes several
millimetres from the site of tracer injection. Use of these tracer techniques
in conjunction with other imaging techniques has revealed that patchy
connections tend to prefer targets with the same response properties. This has
been shown for orientation selectivity and ocular dominance
(Malach et al, 1993)
and CO/interblob zones
(Yoshioka et al, 1996).
The spatial pattern of
these patchy intrinsic connections is therefore closely related to the spatial
pattern of other response-property systems discussed thus far.

Recent work on the postnatal development of area 17 in the Ferret has revealed
that the patchy intrinsic connections form from an initially diffuse set of
random connections in the upper layers
(Ruthazer and Stryker, 1996).
The
formation of patchy connections occurs before the presence of visual input and
before the appearance of response properties associated with the supragranular
layers, for example orientation selectivity. We therefore assume these
connections reflect some primary transformation of early visual inputs, upon
which later fine tuning of response properties is based. This paper describes
two mappings of visual inputs through the granular and supragranular layers,
and how the interaction of the two mappings drives the formation of patchy
intrinsic connections in the supragranular layers.

It has recently been shown, in the primary visual cortex of the tree shrew,
that the patchy intrinsic connections are not perfectly radial, but form an
elongated pattern
(Bosking et al, 1997). The axis of elongation corresponds to
the preferred orientation of the injection site. In other words, if tracer is
injected into a site in the supragranular layers with a preferred orientation
of ,
the pattern of patchy intrinsic connections overlies a
global, retinotopic representation of a line passing through that point and
having an orientation .

4 Local-Global Similar mapping

Alexander et al (1997) have noted strong analogies between the global
properties of the hemi-retinal image and the response properties in the
supragranular layers. These are given in
table 1.
In particular, we assume
these analogies with the hemi-retinal image apply to a geometrical unit in the
supragranular layers corresponding to 1/4 of a hypercolumn. Such a unit
includes one CO blob and one singularity and has the approximate dimensions in
the macaque of 400 um x 300 um (Blasdel, 1992).

Table 1 Analogies between the properties of the hemi-retinal image
and the local organisation of receptive field properties in the supragranular
layers. CO blobs are assumed to be the central visual field's representation in
the local maps of the supragranular layers.

The present model of V1 introduces a few simplifications into the visual
system, and will not explicitly deal with ocular dominance, the cortical
magnification factor, spatial frequency, contrast and colour selectivity. The
resulting simplified model focuses on the development of orientation
selectivity, and aims to explain data from mammals such as the macaque, the
tree shrew and the ferret. A more complete model is in development.

The global retinotopic input-mapping from the hemi-retinal image to the
granular layers (in particular layer 4C) can be approximated by the function of
a complex variable supplied by Schwartz (1980):

1

where

2

and and
take the values 0.333 and 6.66 respectively. This function has
a close fit to the mapping found in the macaque primary visual cortex
(Schwartz, 1983).
The mapping described by
equation 1
is illustrated in
diagram 1.
A semi-circle on the complex plane z, is mapped to an almond shape
on the complex plane, G(z). Since the mapping into the granular
layers is variable across species, to approximate the mapping into the granular
layers we could equally choose

3

or

4

where <= 1.
In the case of the tree shrew, the retinotopic input-mapping
into the granular layer involves little distortion of the hemi-retinal image
(Bosking et al, 1997)
and so could be approximated by an identity mapping. The
model does not depend on the exact mapping chosen for the granular layers.

The retinotopic representation of lamina 4C most directly reaches the
supragranular layers via lamina 4C
and 4A. The LGS model requires that
the influence of the retinotopic representation on the supragranular layers is
at a rather coarse resolution, similar to the blocks of inputs arriving in
layer 4C. We therefore assume that a `pixel' of retinotopic input to the
supragranular layers, ~ 400 x 300 um, serves to drive the formation of local
maps, of the same dimensions, in the supragranular layers.

Receptive field mapping has shown that the retinotopic receptive field
structure of the supragranular layers is not as exact at that in lamina 4C.
This fits with anatomical data suggesting somewhat diffuse inputs into the
supragranular layers via 4A and larger receptive fields for the upper layers
(Blasdel and Fitzpatrick, 1984).
For the sake of simplification, in this
description of the LGS model of V1, we assume that the retinotopic
input-mapping between lamina 4C
and the supragranular layers preserves
perfectly the retinotopic relations.

A second stream of inputs reaches the supragranular layers via laminae
4C and
4B. For simplicity of expression, we describe the local input-mapping to the
supragranular layers in terms of the hemi-retinal visual field, rather than the
representations described in
equations 1-4.
We assume the local input-mapping
to the supragranular layers can be approximated by the mapping of a complex
variable

5

where

6

This mapping may also be expressed as

7

where M is a scaling factor which shrinks the size of the map
(M >> 1). This mapping has the effect of mapping a semi-circle to
a full circle. In the case of n=0, it maps equally spaced lines of
iso-eccentricity to radii that increase as a square of the eccentricity. If
n=1, equally spaced lines of iso-eccentricity are preserved. If
n=2, the pattern described for n=0 is reversed. The mapping
described in
equation 5
is illustrated in
diagram 2.
A set of rays--of angles spanning 180o
--converging on the fovea, is mapped by this function to a
circular pinwheel. Multiple copies of this input-mapping tile the supragranular
layers. According to Schwartz (1980) there are approximately 3,000 hypercolumns
in the macaque primary visual cortex. This leads to an estimate of 12,000
tilings of the local mapping described by
equation 5.

The most critical feature of the mapping is that it is a local mapping.
It allows each retinotopic point to be associated with a local representation
of the whole visual field. The next most important feature of this mapping is
that it maps a semi-circle to a circle, doubling the angles. This approximates,
in the local mappings, the shape of experimentally imaged singularities.

Anatomically, we propose the local input-mapping described by
equations 5-7
occurs through the lateral connections formed by spiny stellates in lamina 4B
(Rockland and Lund, 1983).
These lateral connections recombine the retinotopic
image in a series of steps to form the desired local input-mapping to the
supragranular layers. This is consistent with the observations that 1) the
longest lateral fibres are in 4B, and 2) most fibres within 4B terminate at
great distances within 4B. This anatomy is consistent with the required one to
all mapping. The relevant anatomy is summarised in
diagram 3.
The critical
prediction of the LGS model is that the lateral connections in lamina 4B
preserve the topology of the retinal image, while replicating that topology
many-fold over the surface of V1.

Each point on the supragranular layers can be defined in terms of a
doubly-similar coordinate geometry. Each point is a function of the pair
(zG,zL) where zG
specifies, in discrete complex coordinates, which local input-map the point
falls within, where

8

If there are M tilings of the local input-map, then

9

since the rounded shape of the primary visual cortex means it does not fill the
complex plane of zG entirely. The term zL
specifies the coordinates of the point within that local input-map. Since this
point is a target neuron, zL is also in discrete complex
coordinates. The definition of the point
(zG,zL) in the supragranular layers is
illustrated in
diagram 4.

We express the inverse of the global input-mapping function (e.g. the inverse
of
equation 1)
as G-1(z) and the inverse of the local
input-mapping function (e.g. the inverse of
equation 5) as
L-1(z). Each point,
(zG,zL), in the supragranular layer
receives input from two points in the visual field, one through the global
retinotopic input-mapping, point g:

10

and another through the local input-mapping, point l:

11

where g and l are complex variables and are points on the
hemi-retina. The input into a given point of the supragranular layers, I
(zG,zL), is simply

12

where Ig is the total afferent input from point g and
Il is the total afferent input from point l.

5 Interaction Between Maps

The global input-mapping described by
equations 1-4
supplies the retinotopic
receptive field properties of the supragranular layers. Other response
properties of the local map representations are formed through an interaction
of the local input-maps with the global retinotopic input-mapping into the
supragranular layers. Here we distinguish between the local input-map,
supplied by
equations 5-7
and the local representational maps (or local
response property maps) which form as a result of interactions between the
input maps, other cortical anatomy and hebbian learning driven by visual
stimuli.

The response properties of local representational maps form through hebbian
learning mechanisms as a result of correlated neuronal activity. The LGS
mapping allows each coarse-grained (~ 400 x 300 um) retinotopic input into the
supragranular layers to become associated with a local mapping (also ~ 400 x
300 um) of the entire hemi-field. Since these maps are hypothesised to form
only when both sets of inputs (i.e. Ig and
Il in
equation 12)
are active, and even adjacent local
maps receive different Ig's, each local representational
map in the supragranular layers learns a different set of patterns.

In the general case, each local map in the supragranular layer `sees' a
different set of stimuli. For each Ig, the supragranular
layer also receives inputs, Il, from every point in the
visual hemi-field. Responses to these local mappings of the visual field are
reinforced by activity from the retinotopic inputs Ig.
Only those points in the visual field whose activity reliably coincides with
the point g will form strong connections within the map driven by
Ig. The local representation which forms will be a
version of the visual field which is visually relevant to the
Ig supplying retinotopic input to that local map.

In the case of line orientation preference, each local map `sees' only the
subset of line orientations relevant to that local map. The subset of line
orientations which pass through the hemi-retinal point, g, comprises all
the oriented lines which produce supragranular activity in the local map driven
by that Ig. The local map, insofar as its response
properties are due to the interaction of Ig and
Il, will therefore learn responses to only the subset of
oriented lines passing through the point g. Each local map learns a
subtly different set of lines orientations, depending on its retinotopic
location. The conjunction of the local mapping of an oriented line, and the
global mapping of the point it passes through, is termed a point/orientation
conjunction. A point/orientation conjunction describes the relevant pattern of
activity, or representation, for line orientation preference.

A critical feature of the input-mapping given in
equations 5-7 is that it
simultaneously performs both a contraction of the layer 4C representation to
multiple local images in the supragranular layers and an expansion of the layer
4C representation so that any point in the hemi-field becomes spread out in the
upper layer representation. That is, each local input,
Il, reaches every version of the local map. This
property leads to an explanation for the formation of patchy intrinsic
connections in the supragranular layers.

We assume that the transform described in
equations 5-7
occurs from at least
from the time of early post-natal development, that is, prior to earliest
visual experience. We further assume that patchy connections in the upper
layers are formed from an initial random connectivity. These assumptions are
consistent with recent findings on the development of lateral connections in
the ferret primary visual cortex (Ruthazer and Stryker, 1996).

The lateral connections in the supragranular layers are refined through the
detection of correlated activity, between adjacent local maps, in neurons
receiving inputs from both local and global input-maps. This is illustrated in
diagram 5.
A short line is represented in both the global mapping of the
supragranular layers and the local mappings which tile the supragranular
layers. Where activity in the two adjacent local-map representations coincides,
patchy connections form. Connections are assumed to form between any two of
these representations which have correlated activity over time through a simple
hebbian learning mechanism.

It should be noted that patchy connections form between cortical points in the
supragranular layers (i.e. between Il's) whose
representations are spatially contiguous in layer 4C. The patchy connectivity
therefore reflects the expansion of the cortical image implied by the multiple
tilings of the map described by
equations 5-7.
Patchy connectivity maintains
direct connectivity, within the local supragranular representation, between
representations drawn from the adjacent (or the same) retinotopic locations in
layer 4C.

We propose that the mapping in
equations 5-7
is critical to the development of
lateral connections and also critical to their maintenance over time. However,
once the response properties are acquired during this development process,
moment to moment visual processing in V1 depends primarily on the interactions
between the retinotopic representation in layer 4C and the supra-granular
response properties supplied by the patchy intrinsic connections. If this is
the case then visual processing V1 in is not critically slowed by the proposed
poly-synaptic junctions in the transformation of the representation from layer
4C, through lamina 4B, into the supragranular layers.

The mappings described in this paper reveal how the primary visual cortex
recognises oriented lines in different retinotopic locations. The LGS mapping
uniquely identifies any line by position and angle, since for a given
orientation, there is only one line which passes though that retinotopic
position. The patchy intrinsic connections link together the set of
point/orientation conjunctions, each of which individually defines the line.
The patchy connectivity therefore allows a further avenue of disambiguation,
possibly adding a binding label through phase synchronisation of the neuronal
activity
(Singer, 1993).
Synchronised activity between linked point/orientation
conjunctions enhances response properties to oriented lines, thus generalising
the response property to regions beyond a particular retinotopic location.

In the general case, the model predicts that V1 can learn stimulus features
which are predictable on the basis of spatial contiguity. The interaction, in
the supragranular layers, between a specific retinotopic input and the local
map inputs from the entire visual hemi-field leads to a unique pattern of
activation at that retinotopic point. Properties such as line orientation,
spatial frequency, contrast and colour tend to be spatial contiguous and so
will reliably activate particular patterns of activity in adjacent local
representations. Through this reliable co-activation, hebbian learning drives
the formation of lateral connectivity between the active sites in these
adjacent maps, producing generalised response properties.

A few additional assumptions are required to reproduce the geometric pattern of
orientation preference, at the scale of individual singularities, hypercolumns
and beyond. To retrieve the typical singularity shape, of 180o of
orientation preference traversing 360o
of a circle, responses to oriented
lines of a local map must favour orientations within a range of approximately
180o.
This is the distortion provided by
equation 5.
Systematic mapping of
direction preference in the primary visual cortex of the ferret reveals that
direction preference is most often segregated into local maps with only a
180o range of direction preference (Weliky et al, 1996).

Imaging of orientation preference shows that adjacent singularities are often
mirror images of each other; this produces the saddle regions between sets of
singularities. The tiling of multiple singularities shown in
diagram 6
illustrates this mirroring, revealing multiple saddle points. Reflection of the
adjacent tiles has the effect of allowing orientation preference to change
continuously between singularities. It does not alter the functionality of the
LGS mapping, which only relies on a local representation of the visual
hemi-field being mapped to each global retinotopic point.

Diagram 7
illustrates the relationships between CO blobs and singularities in
the model. The model predicts that singularities and CO blobs should coincide
in the region on the global retinotopic map which represents the centre of the
field of vision. This is because the CO blobs are hypothesised to be the local
map representations of the central field of vision and local-mappings in this
region are activated by the set of orientations passing through the central
field of vision. This contrasts to the local-map representations in the
periphery. The set of orientations passing through these retinotopic locations
miss the central field of vision and hence, in their local map representations,
the CO blobs. This may explain the variation in findings in the literature,
regarding the positioning of CO blobs and singularities
(Bartfeld and Grinvald,
1992; Livingstone and Hubel, 1984).

A simplification in this analysis of orientation preference has been the nature
of the retinotopic inputs into the supragranular layer-hypothesized to be
connections from 4C
via 4A. In reality these connections are diffuse,
rather than point-like. Local map representations have been described in this
paper as though the representation which forms is that one which is
`predictive' for the global retinotopic location over which it lies. It seems
more likely that local representational maps form through the interaction of
the intracortical loops between the granular, supragranular and infragranular
layers
(Miller, 1996).
This would provide the model with additional degrees of
freedom for targets of retinotopic inputs to the supragranular layers. Both the
local input-mapping and the global input-mapping would then be subject to
modification through hebbian learning, depending on how well they predict
activity in the loop through the granular and infragranular layers. This
additional complexity suggests a source for the variation found empirically in
local maps; they are rather more `noisy' than the perfect geometry of the LGS
model would suggest.

6 Conclusions

This paper introduces three assumptions to explain orientation selectivity in
the primary visual cortex. These are: a global, topology-preserving mapping
from the retina to the supragranular layers via the laminae
4C and 4A; a
local, topology-preserving mapping from the retina to the supragranular layers
via laminae 4C
and 4B; and, the formation of patchy intrinsic connections and
orientation preference response properties through correlated activity in the
two mappings.

Three additional assumptions allow the model to predict the specific
geometrical patterns of response properties for orientation preference. These
assumptions are: that the local input-maps distort the hemi-retinal field so
that polar angles are doubled; orientation responses are limited to a range of
180o
within a particular local map; and, the tiling of the supragranular
layers requires two reflections of the local input map. These assumptions allow
the model to reproduce many aspects of the distinct geometry of orientation
preference in the primary visual cortex.

The LGS mapping of visual inputs allows those properties of the visual field
which can be predicted on the basis of spatial contiguity (orientation,
texture, colour, contrast) to be available as response properties over the
entire visual field. The local mapping to the supragranular layers makes the
contents of the entire visual field available for association with each
retinotopic location in the supragranular layer. Patchy intrinsic connections
link these response properties at different retinotopic locations. Further
description of the model will treat direction preference, as well as
simulations of the formation of patchy intrinsic connections
(Alexander et al, ms. in prep.).
Extension of the model to response properties such as colour
selectivity and spatial frequency preference follow from the general case in a
straight forward manner.

The LGS mapping described by
equations 5-7
bears correspondences to the
ice-cube model of Hubel and Wiesel (1977). Both assume the uppers layers are
tiled with a regular map of the feature space. The LGS mapping has the
additional features that it 1) directly suggests the function of known
anatomical connections, 2) explains how the response properties are created.

The model assumes that the geometry of V1 serves to organise multiple response
properties into two dimensions. It differs from other models of V1 geometry
which, rather than mappings, use simulated annealing or other relaxation
procedures to achieve this dimension reduction
(Durbin and Mitchison, 1990; Swindale, 1992).
The latter models have multiple free parameters
(Swindale, 1996).
By contrast, the LGS model of V1 contains a limited number of
assumptions and no arbitrary parameters.

Global mapping of the hemi-retinal image
into the supragranular layer of the primary visual cortex. This mapping is
given in
equation 1.
Left of figure shows the hemi-retinal field which is
mapped to the supragranular layers, shown in the right of figure. The
distortion of the hemi-retinal image is shown by black lines. The point f,
where the converging black lines meet, is the fovea and has the z coordinates
(0,0). The mapping is global, that is, it covers in a single mapping the entire
area of V1. The mapping of a set of oriented lines is shown by coloured lines.
Each coloured line represents the set of moving lines, of a given orientation
and of variable speed, which converge on the point g. In this representation,
the distance of a moving bar from the point g is proportion to its speed.

Local mapping of the hemi-retinal image into the
supragranular layers of the primary visual cortex. This mapping is given in
equation 1,
n=0. Left of figure shows the hemi-retinal field which is mapped to
the supragranular layers, shown at right of figure. The distortion of the
hemi-retinal image is shown by black lines. The point f, where the converging
black lines meet, is the fovea and has the z coordinates (0,0). The mapping is
local, that is, it tiles the area of V1 with multiple small copies of the
z-plane. The mapping of a set of oriented lines is shown by coloured lines.
Each coloured line represents the set of moving lines, of a given orientation
and of variable speed, which converge on the point l. In this representation,
the distance of a moving bar from the point l is proportion to its speed.

Anatomy of V1 relevant to the LGS model. Two
streams of inputs reach the supragranular layers. One mapping, via laminae
4C
and 4A, supplies a global retinotopic mapping. A second stream of
inputs, via 4C
and 4A, supplies a local mapping of the visual
hemi-field.

Schematic representation of the formation of
intrinsic patchy connectivity. A short, oriented line (shown in red) is mapped
according to the LGS model of V1. The left hand column shows the mapping from
hemi-retina to the granular and supragranular layers. Right of diagram shows a
magnified section of the line and the intersecting local maps. Each
point/orientation conjunction uniquely defines the line by position and angle.
Patchy connectivity forms between pairs of the point/orientation conjunctions,
through a hebbian learning rule (shown by black arrows).

Tiling of the supragranular layers with multiple
copies of the hemi-retinal image. Adjacent tilings are reflected about either a
vertically or horizontally oriented axis. The tiling shows saddle points
between singularities.

Relationships between singularities (colour) and
CO blobs (grey).
A. A set of orientations which pass through the centre of the fovea will
create a local supragranular map in which the singularity and CO blob
coincide.
B. A set of orientations which miss the fovea will create a local
supragranular map in which the singularity and CO blob do not coincide.